Monotonicity Formulas for Capillary Surfaces

TL;DR

This paper develops monotonicity formulas for capillary surfaces in specific domains, leading to new inequalities and area estimates that extend previous results to the capillary setting.

Contribution

It extends monotonicity formulas and optimal area estimates to capillary surfaces, broadening the understanding of their geometric properties.

Findings

Established monotonicity formulas for capillary surfaces in half-space and ball.

Derived Li-Yau-type inequalities for Willmore energy of capillary surfaces.

Extended Fraser-Schoen's area estimate to capillary surfaces in the ball.

Abstract

In this paper, we establish monotonicity formulas for capillary surfaces in the half-space $\mathbb{R}^3_+$ and in the unit ball $\mathbb{B}^3$ and extend the result of Volkmann (Comm. Anal. Geom.24(2016), no.1, 195~221. \href{https://doi.org/10.4310/CAG.2016.v24.n1.a7}{https://doi.org/10.4310/CAG.2016.v24.n1.a7}) for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in $\mathbb{B}^3$ (Adv. Math.226(2011), no.5, 4011~4030. \href{https://doi.org/10.1016/j.aim.2010.11.007}{https://doi.org/10.1016/j.aim.2010.11.007}) to the capillary setting, which is different to another optimal area estimate proved by Brendle (Ann. Fac. Sci. Toulouse Math. (6)32(2023), no.1, 179~201. \href{https://doi.org/10.5802/afst.1734}{https://doi.org/10.5802/afst.1734}).